Nearest Neighbor Descent (Dong, Moses, and Li
2011) (NND) is the main way to construct a k-nearest neighbors
graph in `rnndescent`

. Here’s a brief description of the
method.

The idea behind NND is to start with an initial guess of the graph
(typically randomly chosen neighbors) and then iteratively improving
that guess by taking candidate neighbors which are neighbors of
neighbors. For example: if an item `i`

has a current neighbor
`j`

, then `j`

’s neighbors are candidates for
`i`

’s neighbors. The “descent” part is in analogy with
gradient descent, where you can see the sum of the distances in the
graph as an objective function: as better neighbors enter the graph, the
distances must get smaller.

Conceptually it would seem that you would implement this algorithm with a loop like the following in each iteration:

- For each item
`i`

in the graph: - For each item
`j`

in the neighbors of`i`

: - For each item
`k`

in the neighbors of`j`

: - If
`k`

is not already a neighbor of`i`

: - Calculate the distance between
`i`

and`k`

, \(d_{ik}\). - If \(d_{ik}\) is smaller than the
neighbor with the largest distance in the neighbor list of \(i\), update the neighbor list of
`i`

with`k`

.

The process described above involves a lot of looping and repeated
fetching of neighbor vectors, so NND actually uses the concept of a
“local join”. One way to think of it is to consider an item
`i`

fielding requests for its nearest neighbors. It will be
repeatedly asked for it by any other item which considers it a neighbor.
So if we did some work at the start of each iteration to know all the
items which consider `i`

a neighbor, we can generate all the
candidates neighbor pairs that `i`

is involved with at once.
Then we only need to iterate over the items in the graph. We do need to
do the work of finding out who considers `i`

a neighbor but
that also only requires a loop over the graph also.

To be clear, the same amount of work needs to be done, but by doing it in a different order, everything is a bit more efficient in terms of what needs to be fetched from memory.

The up-shot of using the local join approach is that rather than
iterating over the graph one item at a time, we end up a list of pairs
of items `(i, j)`

to update the graph as a whole with. And
because we are dealing with a kNN graph if we have a pair
`(i, j)`

we also have `(j, i)`

as a potential
update, at the cost of only one distance calculation. This has some
challenges in terms of parallel implementation and it also makes caching
distances a bit harder but it’s still better than the more naive
approach of explicitly looping over all neighbors-of-neighbors.

Additionally, there are two other heuristics used to reduce the amount of work done. The first is that candidate neighbors are split into “new” and “old” candidates. A “new” candidate neighbor is any neighbor which entered the graph in the previous iteration. “Old” neighbors are everything else. For the local join, all possible pairs of “new” neighbors are used for updating the graph, but “old” neighbors are only ever paired with “new” neighbors, not other “old” neighbors. This is referred to as “incremental search” in the NND paper.

Also, a tolerance \(\delta\) is used to determine as an early stopping criterion. The total number of items in the graph is \(kN\) where \(k\) is the number of neighbors and \(N\) is the number of items. During each iteration, a counter is incremented every time the graph is successfully updated. If at the end of the iteration the number of updates is less than \(\delta kN\) then the iteration stops.

There is one other minor change to how PyNNDescent works versus the
description in the NND paper, which `rnndescent`

also uses,
which is how sampling of candidates works. For the local join, we need
to know not just the neighbors of `i`

, but those items which
consider `i`

a neighbor, which we call the “reverse
neighbors” of `i`

. While there are always only \(k\) “forward” neighbors of `i`

in a graph, we don’t control who is a neighbor of what, so
`i`

could be the neighbor of many (or even all) the other
items in a dataset. Thus, building the reverse list can be a bit
challenging as we need to be prepared for any item to have up to \(N\) neighbors. In the NND paper, this is
avoided by defining a sample rate \(\rho\), which is used to sample from the
k-nearest neighbors, and then the reverse neighbor list is only built
from the sampled items. A subsequent down-sampling is then applied to
the reverse neighbor list so that both the forward and reverse neighbor
list only contain \(\rho k\) items.

Instead of a sample rate, `rnndescent`

defines a
`max_candidates`

parameter determines the size of both the
forward and reverse neighbor lists per item. If there are more
candidates than the `max_candidates`

value, the retained
candidates are chosen randomly so this works like random sampling.

Finally, instead of a random initialization, PyNNDescent uses a
k-nearest neighbors graph from a random projection forest. There is an
entire vignette explaining how RP forest works. This is also an option
in `rnndescent`

.

It’s easy enough to run NND on a dataset. Here’s an example using the
`iris`

dataset:

The contents of `iris_knn`

is a list with two elements,
both \(N\) by \(k\) matrices where \(N\) is the number of items in the dataset
and \(k\) is the number of neighbors:
`idx`

contains the indices of the neighbors:

and `dist`

contains the distances:

```
iris_knn$dist[1:2, 1:5]
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 0.1000000 0.1414212 0.1414212 0.1414213
#> [2,] 0 0.1414213 0.1414213 0.1414213 0.1732050
```

Apart from `k`

, there are some parameters you may want to
modify:

`metric`

is the distance metric to use. The default is Euclidean distance. There are several metrics you can use. See the documentation for`nnd_knn`

for the full list.`init`

is the initialization method. The default is`"rand"`

which initializes the neighbors randomly. You may wish to use`"tree"`

which uses a random projection forest to initialize the neighbors, similar to`rpf_build`

. To control the tree building, you can pass the same sort of parameters that you would to`rpf_build`

via the`init_args`

parameter. See the vignette on RP forest for more details. You can also pass in a neighbor graph directly. This should have the same format as the output of`nnd_knn`

, i.e. a list of two matrices of size \(N\) by \(k\). NND can be used to refine an existing graph generated by other methods, e.g. RcppAnnoy or RcppHNSW.`n_iters`

is the number of iterations of NND to carry out. The default is to choose based on \(N\), the number of items in the dataset. The amount of work done per iteration decreases quite rapidly, so sticking with the default is usually sensible, especially if you don’t change the convergence criterion`delta`

(see below), because this often causes the algorithm to stop early anyway.`delta`

controls early stopping and must be a value between`0`

and`1`

. If in a given iteration, the number of changes to the neighbor graph is less than`delta * k * N`

then the algorithm stops. The default is`0.001`

so you can interpret that roughly as the neighbor graph needs to have changed by 0.1% to avoid early stopping.`max_candidates`

controls the size of the forward and reverse neighbor lists. The default is to set this to whatever is smaller,`k`

or`60`

.`n_threads`

controls the number of threads to use. The default is to run as a single thread. The slow part of any approximate nearest neighbor algorithm is the distance calculation so using multiple threads is usually a good idea.`ret_forest`

if`TRUE`

, and you have set`init = "tree"`

, then the random projection forest used to initialize the neighbor graph is returned as well. If you want to generate new neighbors based on the original data you will want this.`verbose`

set to`TRUE`

to get information about the progress of the NND.`progress`

this affects how the progress of NND is displayed when`verbose = TRUE`

. The default`bar`

shows a textual progress bar. You can also set`progress = "dist"`

to show the current value of the convergence criterion and the sum of the distances at each iteration. This can help a bit to determine if more iterations or a different convergence criterion will help.

Note that NND uses random number generation to determine the order of
processing candidates, so for reproducible results you should set the
random number seed explicitly. Also, the way that parallelism is
implemented means that reproducibility is not possible for different
settings of `n_threads`

even with a consistent seed,
e.g. going from `n_threads = 0`

to `n_threads = 4`

will give you different results, even if you `set.seed`

with
a fixed seed beforehand.

If you have reason to believe you aren’t getting the results out of
NND that are sufficiently accurate, probably the best thing to do is to
increase `max_candidates`

. Reducing `delta`

or
increasing `n_iters`

usually has less effect. Restarting
`nnd_knn`

with `init`

set to the output of your
previous run usually also helps, but is not a very time-efficient way to
improve matters.

Here is some (lightly edited) sample output when running

`iris_knn <- nnd_knn(iris, k = 15, verbose = TRUE, progress = "dist")`

```
Running nearest neighbor descent for 7 iterations
1 / 7
heap sum = 647.85 num_updates = 3356 tol = 2.25
2 / 7
heap sum = 599.9 num_updates = 216 tol = 2.25
3 / 7
heap sum = 599.9 num_updates = 0 tol = 2.25
Convergence: c = 0 tol = 2.25
```

This tells you that for a dataset of the size of `iris`

,
at most 7 iterations will run. The `1 / 7`

,
`2 / 7`

and so on is logged at the end of each iteration.
Following that is the sum of the distances of the neighbors in the heap,
the number of updates to the neighbor graph and the convergence
criterion. If `num_updates`

falls below `tol`

then
the algorithm stops. In this case, on the third iteration there were no
updates at all, so the algorithm stopped early.

In this case, almost certainly NND has found the exact nearest
neighbors, so you wouldn’t be worried about modifying the parameters.
But if you were so inclined, the output shows you that there would be
little point in increasing `n_iters`

or reducing
`delta`

. This really only leaves `max_candidates`

as an option.

The vignette on dealing with hubness
(where this can be an issue) goes into a bit more detail on how to use
different functions in `rnndescent`

to deal with this sort of
problem.

You can’t. NND can only produce the k-nearest neighbors graph for the provided data. It doesn’t produce an “index” of any kind that you can query. The value of NND and the local join really only makes sense if you can take advantage of the fact that calculating the distance \(d_{ij}\) lets update the neighbor list of \(i\) and \(j\) at once.

If you try to apply the concepts from NND to querying new data you
quickly end up at a method that looks a lot like most greedy graph-based
searches. For that, you should look at `graph_knn_query()`

,
although as noted above you can also use the random projection forest
used to initialize the neighbor graph when `init = "tree"`

.
You will also probably want to augment the neighbor graph to make it
more amenable for searching using
`prepare_search_graph()`

.

Dong, Wei, Charikar Moses, and Kai Li. 2011. “Efficient k-Nearest
Neighbor Graph Construction for Generic Similarity Measures.” In
*Proceedings of the 20th International Conference on World Wide
Web*, 577–86.